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arXiv:1107.1472 · Physical Review A

Viscosity sum rules at large scattering lengths

Walter D. Goldberger, Zuhair U. Khandker

Published Jan 18, 2012• 22 citations
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Abstract

We use the operator product expansion (OPE) and dispersion relations to obtain model-independent ``Borel-resummed'' sum rules for both shear and bulk viscosity of many-body systems of spin-1/2 fermions with predominantly short-range $S$-wave interactions. These sum rules relate Gaussian weights of the frequency-dependent viscosities to the Tan contact parameter $\mathcal{C}(a)$. Our results are valid for arbitrary values of the scattering length $a$, but receive small corrections from operators of dimension $\ensuremath{\Delta}>5$ in the OPE and can be used to study transport properties in the vicinity of the $a\ensuremath{\rightarrow}\ensuremath{\infty}$ fixed point. In particular, we find that the exact dependence of the shear viscosity sum rule on scattering length is controlled by the function $\mathcal{C}(a)$. The sum rules that we obtain depend on a frequency scale ${\ensuremath{\omega}}_{0}$ that can be optimized to maximize their overlap with low-energy data.

DOI
10.1103/physreva.85.013624

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We use the operator product expansion (OPE) and dispersion relations to obtain model-independent ``Borel-resummed'' sum rules for both shear and bulk viscosity of many-body systems of spin-1/2 fermions with predominantly short-range SSS-wave interactions. These sum rules relate Gaussian weights of the frequency-dependent viscosities to the Tan contact parameter C(a)\mathcal{C}(a)C(a). Our results are valid for arbitrary values of the scattering length aaa, but receive small corrections from operators of dimension \ensuremathΔ>5\ensuremath{\Delta}>5\ensuremathΔ>5 in the OPE and can be used to study transport properties in the vicinity of the a\ensuremath→\ensuremath∞a\ensuremath{\rightarrow}\ensuremath{\infty}a\ensuremath→\ensuremath∞ fixed point. In particular, we find that the exact dependence of the shear viscosity sum rule on scattering length is controlled by the function C(a)\mathcal{C}(a)C(a). The sum rules that we obtain depend on a frequency scale \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ that can be optimized to maximize their overlap with low-energy data.
We use a dispersion relation in conjunction with the operator product expansion (OPE) to derive model independent sum rules for the dynamic structure functions of systems with large scattering lengths. We present an explicit sum rule for the structure functions that control the density and spin response of the many-body ground state. Our methods are general and apply to either fermions or bosons which interact through two-body contact interactions with large scattering lengths. By employing a Borel transform of the OPE, the relevant integrals are weighted toward infrared frequencies, thus allowing for greater overlap low energy data. Similar sum rules can be derived for other response functions. The sum rules can be used to extract the contact parameter introduced by Tan, including universality violating corrections at finite scattering lengths.
The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of nonrelativistic quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$, respectively, to derive exact, nonperturbative results. Our results include a microscopic connection between the shear viscosity $\ensuremath{\eta}$ and the normal-fluid density ${\ensuremath{\rho}}_{n}$; sum rules for $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$ and their evolution through the BCS-BEC crossover (where BEC denotes Bose-Einstein condensate); and universal high-frequency tails for $\ensuremath{\eta}(\ensuremath{\omega})$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$. We use our sum rules to show that, at unitarity, $\ensuremath{\zeta}(\ensuremath{\omega})$ is identically zero and thus relate $\ensuremath{\eta}(\ensuremath{\omega})$ to density-density correlations. We predict that frequency-dependent shear viscosity $\ensuremath{\eta}(\ensuremath{\omega})$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.
We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, nnn is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.
We consider a homogeneous, balanced gas of strongly interacting fermions in two spin states interacting through a large scattering length. Finite-range corrections are needed for a quantitative description of data which experiments and numerical simulations have provided. We use a perturbative field-theoretical framework and a tool called the operator product expansion (OPE), which together allow for the expression of finite-range corrections to the universal relations and momentum distribution. By using the OPE, we derive the 1/k61/{k}^{6}1/k6 part of the momentum tail, which is related to the sum of the derivative of the energy with respect to the finite range and the averaged kinetic energy of opposite spin pairs. By comparing the 1/k41/{k}^{4}1/k4 term and the 1/k61/{k}^{6}1/k6 correction in the momentum distribution to provided quantum Monte Carlo (QMC) data, we show that including the 1/k61/{k}^{6}1/k6 part offers marked improvements. Our field-theoretical approach allows for a clear understanding of the role of the scattering length and finite effective range in the universal relations and the momentum distribution.
We compute the shear viscosity, thermal conductivity, and spin diffusivity of a Fermi gas with short-range interactions in the Fermi liquid regime of the normal phase, that is, at temperatures T much lower than the Fermi temperature T_{F} and larger than the superfluid critical temperature T_{c}. In line with recent advances in the precision of cold atom experiments, we provide exact results up to the second order in the interaction strength. We extend the Landau-Salpeter equation to compute the collision amplitude beyond the forward-scattering limit, covering all collisions on the Fermi surface. We treat the collision kernel exactly, leading to significant corrections beyond relaxation-time or variational approximations. The transport coefficients, as functions of the s wave scattering length a and Fermi wave number k_{F}, follow (1+γk_{F}a)/a^{2} up to corrections of order O(a^{0}), with a positive coefficient γ for the viscosity and negative one for the thermal conductivity and spin diffusivity.
We apply the operator-product expansion to determine the asymptotic form of the current response of a Fermi gas in two and three dimensions. The leading-order term away from the one-particle peak is proportional to a quantity known as the contact, the coefficient of which is determined exactly. We also calculate the dynamic structure factor and the high-frequency tails of the spectral viscosities as a function of the scattering length. Our results are used to derive certain sum rules for the viscosities.